EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-EP-2017-218 2018/10/11
CMS-HIN-15-008
Pseudorapidity and transverse momentum dependence of
flow harmonics in pPb and PbPb collisions
The CMS Collaboration
∗Abstract
Measurements of azimuthal angular correlations are presented for high-multiplicity pPb collisions at√sNN =5.02 TeV and peripheral PbPb collisions at√sNN =2.76 TeV. The data used in this work were collected with the CMS detector at the CERN LHC. Fourier coefficients as functions of transverse momentum and pseudorapidity are studied using the scalar product method, 4-, 6-, and 8-particle cumulants, and the Lee–Yang zeros technique. The influence of event plane decorrelation is evaluated using the scalar product method and found to account for most of the observed pseu-dorapidity dependence.
Published in Physical Review C as doi:10.1103/PhysRevC.98.044902.
c
2018 CERN for the benefit of the CMS Collaboration. CC-BY-4.0 license
∗See Appendix A for the list of collaboration members
1
1
Introduction
High energy density matter with quark and gluon degrees of freedom, a state of matter known as the quark-gluon plasma (QGP), is created in relativistic heavy ion collisions at the BNL RHIC and at the CERN LHC [1–6]. The energy density created in the initial heavy ion collision is az-imuthally nonuniform as a consequence of the collision geometry and its fluctuations. Interac-tions among constituents in the QGP convert this nonuniformity into an observable anisotropy in the final-state particle momentum distribution. The azimuthal angle distribution of emitted particles can be characterized by its Fourier components [7]. In particular, the second and third Fourier components, v2 and v3, known as elliptic and triangular flow, respectively, most
di-rectly reflect the medium response to the initial collision geometry and its fluctuations [8]. The magnitudes of these components provide insights into the fundamental transport properties of the medium [9–11]. Two-particle correlations in the azimuthal angle (φ) and pseudorapidity (η) differences between the two particles (∆φ and ∆η) have played a vital role in the observation of the azimuthal anisotropies [12–19]. These particle correlations are characterized by a pro-nounced structure at|∆φ| ≈0 extending over a large∆η range (referred to as the “ridge”). In collisions between two heavy nuclei, such as CuCu and AuAu collisions at RHIC [12–14] and PbPb collisions at the LHC [16–19], these long-range correlations are often attributed to the collective flow from a strongly interacting, expanding medium [20, 21]. This is corroborated by multiparticle correlations, suggesting a hydrodynamic origin for the observed azimuthal anisotropies [22].
The lightest systems in which ridge-like structures have been observed include high-multiplicity final states in pp [23–27] and pPb [27–32] collisions at the LHC. Evidence of such long-range
correlations is also observed at a nucleon-nucleon center-of-mass energy of√sNN = 200 GeV
in pAu [33], dAu [34–36] and 3HeAu collisions [37] at RHIC. In pPb collisions, the overall
strength of the correlation is observed so far to be significantly larger than in pp collisions, and is comparable to that found in peripheral PbPb collisions [38, 39].
Both the ATLAS [40, 41] and CMS [38] experiments have measured significant elliptic flow coefficients in pPb collisions at√sNN = 5.02 TeV using four-particle correlations based on the cumulant method [42]. The long-range correlations persist in measurements that study the correlation among six or more particles in pPb collisions [26, 39, 43] and in measurements of four-particle and six-particle correlations in pp collisions at√s=13 TeV [26, 41]. Four-particle
correlation measurements in the dAu system at √sNN = 200, 62.5, 39, and 19.6 GeV by the
PHENIX Collaboration and a six-particle correlation measurement by the same collaboration at√sNN =200 GeV also find significant elliptic flow coefficients [44].
In combination, these measurements support a collective origin of the azimuthal correlations, and have raised the possibility that a QGP droplet might be formed in small-system collisions exhibiting fluid-like behavior [28–30, 39, 45]. If such a mechanism can be confirmed, it will significantly extend the range of system size for which the QGP medium is considered to exist. However, the origin of the ridge phenomenon in small collision systems is still being actively investigated. In addition to a hydrodynamic origin [45, 46], possible alternative explanations include gluon saturation in the initial interacting state of the protons [47, 48], multiparton inter-actions [49], and the anisotropic escape of partons from the surface of the interaction region [50]. To provide further constraints on the theoretical understanding of the azimuthal anisotropies in different collision systems, this paper presents results on the pseudorapidity and transverse
momentum dependence of the flow harmonics in pPb and PbPb collisions. The v2 coefficients
are measured using the 4-, 6-, and 8-particle Q-cumulants [51], the Lee–Yang zeros (LYZ) [52], and the scalar product methods [53, 54]. The v3coefficients, which result from fluctuations in
the collision geometry, are studied with the scalar product method. Within the hydrodynamic picture, the longer lifetime of the medium on the Pb-going side in pPb collisions is expected to lead to larger values for both the v2 and v3 flow harmonics than on the p-going side [55].
The pPb system is studied at√sNN = 5.02 TeV using data obtained by the CMS experiment in
2013. A sample of PbPb collision data at√sNN = 2.76 TeV is also analyzed. The particle corre-lations are studied for high-multiplicity pPb collisions whose particle densities are comparable to those in mid-central (50–60% centrality) PbPb collisions. The centrality variable is defined as a fraction of the inelastic hadronic cross section in heavy ion collisions, with 0% corresponding to the most central, i.e., head-on collisions. This allows for a direct comparison of pPb and PbPb systems over a broad range of similar particle multiplicities, thereby helping to clarify the underlying mechanism responsible for the observed correlations.
2
The CMS experiment
A detailed description of the CMS detector can be found in Ref. [56]. The results in this paper are mainly based on the silicon tracker detector and two hadron forward calorimeters (HF) lo-cated on either side of the tracker. Situated inside the 3.8 T field of a super-conducting solenoid, the silicon tracker consists of 1 440 silicon pixel and 15 148 silicon strip detector modules. It measures charged particles within the range of |η| < 2.4 and provides an impact parameter
resolution of≈15 µm and a pTresolution better than 1.5% at pT ≈100 GeV/c. Electromagnetic
(ECAL) and hadron (HCAL) calorimeters are also located inside the solenoid and cover the range of|η| < 3.0. The HCAL has sampling calorimeters composed of brass and scintillator
plates. The ECAL consists of lead-tungstate crystals arranged in a quasi-projective geometry. Iron/quartz-fiber Cherenkov HF cover the range 2.9< |η| < 5.2 on either side of the
interac-tion region. The HF calorimeters, which are used in the scalar product analysis, are azimuthally
subdivided into 20◦ modular wedges and further segmented to form 0.175×10◦ (∆η×∆φ)
towers. The CMS detector response is determined through Monte Carlo (MC) studies using GEANT4 [57].
3
Event and track selection
The pPb data set corresponds to an integrated luminosity of 35 nb−1. The beam energies were 4 TeV for protons and 1.58 TeV per nucleon for lead nuclei, resulting in√sNN = 5.02 TeV. The beam directions were reversed during the run. The results from both beam directions are com-bined using the convention that the proton-going direction defines positive pseudorapdity. As a result of the energy difference between the colliding beams, the nucleon-nucleon center-of-mass frame in the pPb collisions is not at rest with respect to the laboratory frame. Massless particles emitted at ηc.m. = 0 in the nucleon-nucleon center-of-mass frame will be detected at
η=0.465 in the laboratory frame. Unless otherwise stated, all pseudorapidities reported in this
paper are referred to with respect to the laboratory frame. A sample of√sNN = 2.76 TeV PbPb data collected during the 2011 LHC heavy ion run, corresponding to an integrated luminosity of 2.3 µb−1, is also analyzed for comparison purposes. The triggers, event selection, and track reconstruction are identical to those used in Ref. [38].
In order to select high-multiplicity pPb collisions, dedicated high-multiplicity triggers were implemented using the CMS level-1 and high-level trigger (HLT) systems. The online track reconstruction at the HLT is based on the three layers of pixel detectors, and requires a track origin within a cylindrical region of length 30 cm along the beam axis and radius 0.2 cm per-pendicular to the beam axis, centered at the nominal interaction point. For each event, the
3
vertex reconstructed with the highest number of pixel tracks is selected. The number of pixel tracks (Ntrkonline) with|η| <2.4, pT >0.4 GeV/c, and a distance of closest approach to this vertex
of 0.4 cm or less, is determined for each event. Several high-multiplicity ranges are defined with prescale factors that are progressively reduced until, for the highest multiplicity events, no prescaling was applied.
In the offline analysis, hadronic collisions are selected by requiring a coincidence of at least one HF tower containing more than 3 GeV of total energy on either side of the interaction region. Only towers within 3.0< |η| <5.0 are used in order to avoid the edges of the HF acceptance.
The pPb interactions were simulated with both the EPOS LHC [58] and the HIJING 1.383 [59]
event generators. The requirement of having at least one primary particle with total energy E > 3.0 GeV in each of the η ranges −5.0 < η < −3.0 and 3.0 < η < 5.0 is found to select
97–98% of the total inelastic hadronic cross section.
Events in the offline analysis are also required to contain at least one reconstructed primary ver-tex within 15 cm of the nominal interaction point along the beam axis (zvtx) and within 0.15 cm
transverse to the beam trajectory. At least two reconstructed tracks are required to be associ-ated with the primary vertex. Beam-relassoci-ated background is suppressed by rejecting events for which less than 25% of all reconstructed tracks pass the track selection criteria for this analysis. The pPb instantaneous luminosity provided by the LHC in 2013 resulted in an approximately 3% probability of at least one additional interaction occurring in the same bunch crossing. Such pileup events become more significant as the event multiplicity increases. Following the pro-cedure developed in Ref. [38] for rejecting pileup events, a 99.8% purity of single-interaction events is achieved for the pPb collisions belonging to the highest multiplicity class of this anal-ysis.
The CMS “high-quality” tracks described in Ref. [60] are used in this analysis. Additionally, a reconstructed track is only considered as a candidate track from the primary vertex if the significance of the separation along the beam axis (z) between the track and the best vertex, dz/σ(dz), and the significance of the track impact parameter measured transverse to the beam,
dT/σ(dT), are each less than 3. The relative uncertainty in pT, σ(pT)/pT, is required to be less
than 10%. To ensure high tracking efficiency and to reduce the rate of incorrectly reconstructed tracks, only tracks within|η| <2.4 and with pT >0.3 GeV/c are used in the analysis. The entire
pPb data set is divided into classes of reconstructed track multiplicity, Ntrkoffline, where primary tracks with |η| < 2.4 and pT > 0.4 GeV/c are counted. A different pT cutoff of 0.4 GeV/c is
used in the multiplicity determination because of the constraints on the online processing time for the HLT. The multiplicity classification in this analysis is identical to that used in Ref. [38], where more details are provided, including a table relating Ntrkofflineto the fraction of minimum bias triggered events.
The peripheral PbPb data collected during the 2011 LHC heavy ion run with a minimum bias trigger are also reanalyzed in order to compare directly the pPb and PbPb systems in the same Ntrkofflineranges [38]. This PbPb sample is reprocessed using the same event selection and track reconstruction as for the present pPb analysis. A description of the 2011 PbPb data set can be found in Ref. [61]. The correspondence between the PbPb Ntrkofflinevalues and the total energy deposited in the HF [62], as characterized by a collision centrality, is given in Ref. [38], ranging from 67% centrality for Ntrkoffline= 120 to 55% centrality for Ntrkoffline= 300.
4
Analysis
4.1 Scalar product method
In previous publications, CMS has analyzed the elliptic [62] and higher-order [63] flow coeffi-cients for PbPb collisions at√sNN =2.76 TeV using the “traditional” event plane method [64]. It is now known that fluctuations in the participant geometry lead to vncoefficients that can vary
event-by-event, with the average coefficientshvnibeing smaller than the corresponding
root-mean-square values, phv2
ni. The vn values found using the traditional event plane method
will fall somewhere between these two limits [54]. The scalar product method [53, 54], which is used in this paper, avoids this ambiguity and gives results that correspond tophv2
ni[54].
The event plane angles can be expressed in terms of Q-vectors. For a perfect detector response, the Q-vector corresponding to the nth-order azimuthal asymmetry for a given event is defined as ~ Qn = Qnx, Qny = ~ Qn cos(nΨn), ~ Qn sin(nΨn) = M
∑
i=1 wicos(nφi), M∑
i=1 wisin(nφi) ! , (1)where M is the subevent multiplicity, φiis the azimuthal angle of the ith particle, wiare
weight-ing factors, and the correspondweight-ing event plane angle is given as
Ψn= 1 n tan −1 Q ny Qnx . (2)
Different weights wi are possible. For example, the Q-vectors with wi = 1 relate to the
az-imuthal particle density, with wi = pT,i to the transverse momentum distribution, and with
wi = ET,i to the transverse energy distribution. Since the vn(pT)coefficients increase with pT
up to≈3 GeV/c, the choice of either pTor ETweighting generally results in a better event plane
angle resolution than a unity particle weighting [64]. Expressed in terms of complex weighted q-vectors, where
qn= M ∑ i=1 wieinφi W , (3) and W = ∑M i=1
wi, the scalar product coefficients are found with
vn{SP} ≡ hqnq∗nAi r hqnAq∗nBihqnAq∗nCi hqnBq∗nCi . (4)
In Eq. (4), the weighted averagehifor vectors qnαand qnβwith total weights Wαand Wβ, where
αand β correspond to the second subscripts (if present) on the q-vectors in Eq. (4), is given by
D qnαq∗nβ E =Re Nevt ∑ i=1
WαiWβiqnαiq∗nβi
Nevt ∑ i=1 WαiWβi , (5)
4.1 Scalar product method 5
where Nevtis the total number of events. The A, B, and C subscripts in Eq. (4), denoted using α
and β in Eq. (5), refer to pseudorapidity ranges for which event planes are determined. Here, the “reference” event plane is the A plane, and the B and C planes are used to correct for the finite resolution of the A plane. The q-vector with only one subscript, qn in Eq. (4), is based
on tracks within the specific pTand η range for which the azimuthal asymmetry coefficient is
being measured. Unit weights are used in Eq. (1) in this case.
The two HF calorimeters are used to determine the A and B event planes, with the C plane established using the tracker. In the HF detector regions, with 3.0 < |η| < 5.0, the sums in
Eq. (1) are taken over the towers and the weights are taken as the transverse energy deposited in each tower, with no restriction placed on the tower energy. For the tracker-based C plane, the sums are over the individual tracks with 0.3 < pT < 3.0 GeV/c and the weights are taken
as the corresponding pTvalues. The Q-vectors corresponding to event planes A, B, and C are
“recentered” to account for nonuniformities in the detector response [64, 65]. In recentering, the averages over all events of the x- and y-terms in Eq. (1) (hQnxiandQny ) are subtracted
on an event-by-event basis when calculatingQ~Recenteredn . That is, ~
QRecenteredn = Qnx− hQnxi, Qny−Qny . (6)
The value of qn in Eq. (4) is based on tracks within a specific pT and η range for which the
azimuthal asymmetry coefficient is being measured. In this case, unit weights are used in Eq. (1) and no recentering corrections are applied.
It has been noted recently [66–69], and experimentally confirmed by CMS [70], that the event plane angle should not be considered a global event observable. In the CMS study [70], the
decorrelation between the event plane angles at pseudorapidity ηA and ηB is found to follow
the functional form:
cos[2{Ψn(ηB) −Ψn(ηA)}] =e−F
η
n|ηB−ηA|, (7)
where Fnη is the decorrelation strength.
Such a decorrelation can arise from fluctuations of the geometry of the initial-state nucleons and their constituent partons [66–68]. Previously it has been assumed that Fourier coefficients at pseudorapidity ηROI, where ROI stands for “region of interest”, can be deduced using event
plane angles found in a different pseudorapidity range (say, at ηA), with the caveat that a
suf-ficient pseudorapidity gap is present to avoid short-range correlations. The event plane angle found at ηAis viewed as approximating a global participant plane angle set by the initial
colli-sion geometry and only differing from the ideal by its finite resolution, which, in turn, depends
on both the number of particles used to define the angle and the azimuthal asymmetry at ηA.
The event plane resolution is accounted for in Eq. (4) by determining event planes in three separate regions of η and assuming that these planes reflect the same underlying geometry, only differing by their respective resolutions. The variation with pseudorapidity breaks this
assumption and can have a significant effect on the harmonic coefficient values vn deduced
using either the traditional or scalar product methods.
Considering event plane decorrelation, each of the scalar products in Eq. (4) will be reduced by the decorrelation effect as indicated in Eq. (7). If the decorrelation strength Fnremains relatively
constant as a function of the pseudorapidity gap between event planes, the vn{SP}coefficient in
¯vn{SP}with υn{SP} = hqnq∗nAie−Fn|ηA−ηROI| s hqnAq∗nBie −Fn|ηA−ηB|hqnAq∗ nCie −Fn|ηA−ηC| hqnBq∗nCie−Fn|ηB−ηC| = ¯υn{SP} e−Fn|ηA−ηROI| e−12Fn{|ηA−ηB|+|ηA−ηC|−|ηC−ηB|} = ¯υn{SP}e−Fn|ηC−ηROI|, (8)
where ηC is taken to fall between ηAand ηB. Short-range, nonflow correlations, such as
back-to-back dijets, resonance decay, etc., are again suppressed by having a pseudorapidity gap between ηROIand ηA.
For the “standard” analysis using a three subevent resolution correction where both the third
subevent angle (ΨC
n) and the particles belonging to the region of interest are at midrapidity
(ηROI =ηC ≈0), it follows that the decorrelation effect will not strongly influence the deduced
Fourier coefficient vn. It can be noted that the same result is expected if a two-subevent
resolu-tion correcresolu-tion is used, as is commonly done for symmetric collision systems. However, if ηROI
is different from ηC, the deduced vnvalue will be reduced by the decorrelation effect.
The pseudorapidity-dependent decorrelation of event planes can occur through different mech-anisms. Equation (8) assumes a Gaussian decorrelation characterized by a fixed Fnvalue. It is
also possible for Fnη to vary with η, in which case the η dependence shown in Eq. (7) and (8)
would be more complicated. A simplified MC simulation was used to explore the two Gaus-sian spreading scenarios, corresponding to a fixed or η-dependent Fnηfactor. It was found that
the input vnvalues could be recovered by moving theΨCn event plane along with the particles
of interest. An alternative source of decorrelation is the situation where rotation of the event plane angle results from a torque effect rather than a random spreading [67]. In this case, the MC simulations showed that moving theΨCn event plane does not fully correct for the decorre-lation, although it does lead to results closer to the input values than is found by setting ηC =0.
A comparison of the v2and v3results obtained with ηC =0 and with ηC =ηROImight help in
estimating the relative importance of the different types of decorrelation possible in heavy ion collisions. Event plane results using both of these assumptions for ηCare reported.
Two different reference event planes are used in the analysis: HF−(−5.0<η< −3.0)and HF+
(3.0<η<5.0). The corresponding resolution correction factors are determined with the three
subevent method where, for the HF+(HF−) reference plane (A-plane), the resolution correction is based on the HF−(HF+) event plane (B-plane) as well as either the midrapidity tracker event
plane, with−0.8< η< 0.8, or with event planes that correspond to the pseudorapidity range
of the ROI (C-plane). Since analyses where the midrapidity event plane ηC is taken within
−0.8< ηC <0.8 and analyses where ηC =ηROIare both presented, the convention is adopted
of labelling results as “ηC =0” or “ηC =ηROI,” respectively.
4.2 Cumulant method
If the particles emitted in a collision are correlated with a global reference frame, they will also be correlated with each other. The cumulant method explores the collective nature of the anisotropic flow through the multiparticle correlations. As the number of particles in the correlation study increases, the cumulant values will decrease if only part of the particle sample shares a common underlying symmetry, as would be the case for dijets. The flow harmonics
4.2 Cumulant method 7
correlators are first defined by
hh2ii ≡DDein(φ1−φ2)EE, hh4ii ≡DDein(φ1+φ2−φ3−φ4) EE , hh6ii ≡DDein(φ1+φ2+φ3−φ4−φ5−φ6) EE , hh8ii ≡DDein(φ1+φ2+φ3+φ4−φ5−φ6−φ7−φ8) EE , (9)
where φi is the azimuthal angle of the ith particle, and hh. . .ii indicates that the average is
taken over all m-particle combinations for all events. In order to remove self-correlations, it is required that the m particles be distinct. The unbiased estimators of the reference m-particle cumulants [51], cn{m}, are defined as
cn{4} =hh4ii −2hh2ii2,
cn{6} =hh6ii −9hh4iihh2ii +12hh2ii3,
cn{8} =hh8ii −16hh6iihh2ii −18hh4ii2
+144hh4iihh2ii2−144hh2ii4.
(10)
The reference flow v2{m}obtained by correlating the m particles within the reference phase
space of|η| <2.4 and pTrange of 0.3< pT <3.0 GeV/c was presented in Ref. [39] using
vn{4} = 4 q −cn{4} vn{6} = 6 q cn{6}/4, vn{8} = 8 q −cn{8}/33. (11)
The cumulant calculations are done using the code described in Ref. [71].
By replacing one of the particles in a correlator for each term in Eq. (9) with a particle from certain ROI phase space in pT or η, with the corresponding correlators denoted by primes, one
can derive the differential m-particle cumulants as dn{4} =hh40ii −2hh2iihh20ii,
dn{6} =hh60ii −6hh2iihh40ii −3hh20iihh4ii +12hh20iihh2ii2,
dn{8} =hh80ii −12hh2iihh60ii −4hh20iihh6ii
−18hh40iihh4ii +72hh4iihh2iihh20ii +72hh40iihh2ii2−144hh20iihh2ii3.
(12)
Then the differential v2{m}(pT, η)can be extracted as
vn{4}(pT, η) = −dn{4}/(−cn{4})3/4, vn{6}(pT, η) = dn{6} 4 . c n{6} 4 5/6 , vn{8}(pT, η) = −dn{8} 33 .−cn{8} 33 7/8 . (13)
An efficiency weight is applied to each track to account for detector nonuniformity and effi-ciency effects. For this analysis, the work of Ref. [71] was extended to allow for the explicit calculation of the differential Q-cumulants for the first time.
4.3 Lee–Yang zeros method
The LYZ method [52] allows for a direct study of the large-order behavior by using the asymp-totic form of the cumulant expansion to relate locations of the zeros of a generating function to the azimuthal correlations. This method has been employed in previous CMS PbPb and pPb analyses [39, 62, 63]. The v2 harmonic averaged over 0.3 < pT < 3.0 GeV/c is found for each
multiplicity bin using an integral generating function [17]. Similar to the cumulant methods, a weight for each track is implemented to account for detector-related effects. Anisotropic flow is formally equivalent to a first-order phase transition. As a result, the first zero of the generating grand partition function can be viewed as anisotropic flow of the final-state system.
The integrated flow for the harmonic n is the average value of the flow Q-vector projected onto the unit vector with angle nΦR,
vintn ≡Qnxcos(nΦR) +Qnysin(nΦR)
= DQΦR
n
E
, (14)
whereΦRis the actual reaction-plane angle. SinceΦRis not an observable, the LYZ method is
used to obtain an estimate of this quantity. In the present analysis, a complex product generat-ing function is first defined as
Gθ n(ir) = D gθ n(ir) E =D M
∏
j=1 [1+ir wjcos(n(φj−θ))] E , (15)where M is the event multiplicity, φj and wj are, respectively, the azimuthal angle and the
weight of the jth particle, the averagehiis taken over all events, and θ is chosen to take discrete
values within the range [0, π/n)as
θ = k
nθ
π
n, k=0, 1, 2, ..., nθ−1. (16)
The number of projection angles is set to nθ = 5 to get the average values. This number was
found in the previous CMS studies to achieve convergence of the results [39, 62, 63]. To calculate the yield-weighted integral flow, Gθ
nis evaluated for many values of the real
posi-tive variable r. Plotting the modulus|Gθ
n(ir)|as a function of r, the integrated flow is directly
related to the first minimum rθ
0of the distribution, with
vθ,int n {∞} ≡ j01 rθ 0 , (17)
where j01≈2.405 is the first root of the Bessel function J0(x). The quoted results involve a final
average over different θ values, with vintn = 1 nθ nθ−1
∑
θ=0 vθ,int n {∞}. (18)After the integrated flow coefficient vintn is determined, the pT- and η-dependent v2{LYZ}
val-ues are found using
vθ n vθ,int n =Re D gθ(irθ 0) cos(n(φj−θ)) 1+irθ 0wjcos(n(φj−θ)) E φ D gθ(irθ 0)∑j wjcos(n(φj−θ)) 1+irθ 0wjcos(n(φj−θ)) E . (19)
4.4 Systematic uncertainties 9
The average h...iφ in the numerator is taken over the particles in the ROI. The average in the denominator is over all particles with 0.3 < pT < 3.0 GeV/c and |η| < 2.4. Again, the final
results involve an average over the different θ values vn = 1 nθ nθ−1
∑
θ=0 vθ n. (20) 4.4 Systematic uncertaintiesThe systematic uncertainties resulting from the track selection and efficiency, from the vertex position, and from the pileup contamination contribute to all three methods (scalar product, cumulant, and LYZ). The effects of track quality requirements were studied by varying the track selection requirements, dz/σ(dz)and dT/σ(dT), from 2 to 5, and σ(pT)/pT from 5% to
the case where this requirement is not applied. A comparison of the results using efficiency
correction tables fromEPOS andHIJINGMC event generators was made to study the tracking
efficiency uncertainty. By comparing the results from different event primary vertex positions along the beam direction, with|zvtx| <3 cm and 3< |zvtx| <15 cm, it is possible to investigate
the uncertainties coming from the tracking acceptance effects. The effects of pileup events were studied by looking at events where there was only one reconstructed vertex. The experimental systematic effects are found to have no significant dependence on Noffline
trk , pT, or η.
The v2 systematic uncertainties associated with the PbPb collision results were found to be
comparable for the three methods (≈3%), with contributions from the track selection and effi-ciency (1–2%), the vertex position (1–2%), and pileup effects (<1%). Similar uncertainties are found for pPb collisions based on both the cumulant and scalar product methods. For the LYZ pPb results, a more conservative uncertainty of 11% is quoted based on the large statistical uncertainties associated with the corresponding systematic studies.
In addition, a comparison was done between the results for the two different beam directions. For the event plane analysis, the p-side and Pb-side HF detectors used to determine the event plane angles are switched by changing the beam direction. Based on this study, where the small magnitude of the v3coefficient limits the statistical significance of the systematic studies,
a larger, conservative systematic uncertainty is assigned to the v3{SP} results of 10%. The
overall systematic uncertainties are summarized in Table 1, and shown as grey boxes in the figures.
Table 1: Systematic uncertainties. v2(pT) v2(η) v3 Scalar product pPb 3% 3% 10% PbPb 3% 3% 10% Cumulant pPb 3% 3% — PbPb 3% 3% — Lee–Yang zeros pPb 11% 11% — PbPb 3% 3% —
The multiparticle cumulant and LYZ analyses are expected to be relatively insensitive to non-flow effects. For the scalar product method, however, the nonnon-flow effects can become signifi-cant as the differential particle density decreases, as is the situation for the lower Ntrkofflineranges
and for higher pTvalues. Also, the nonflow effects become more significant as the gap between
the primary event plane (ηA) and the region of interest (ηROI) becomes small. In this paper, the
nonflow influence on the scalar product results is viewed as part of the physics being explored and is not taken as a systematic uncertainty.
5
Results
We first explore the transverse momentum dependence of v2 and v3 in pPb and PbPb at
com-parable particle multiplicities. The v2 values were found using the scalar product, m-particle
cumulant, and LYZ methods, denoted as v2{SP}, v2{m}, and v2{LYZ}, respectively, while v3
was found using only the scalar product method.
0 2 4 (GeV/c) T p 0.1 0.2 2 v < 150 offline trk N ≤ 120 2 4 (GeV/c) T p 0.1 0.2 < 185 offline trk N ≤ 150 2 4 (GeV/c) T p 0.1 0.2 < 220 offline trk N ≤ 185 2 4 (GeV/c) T p 0.1 0.2 < 260 offline trk N ≤ 220 CMS 35 nb-1 (pPb 5.02 TeV) | < 2.4 η | | > 2} η ∆ {2,| 2 v | < 2.4 η | {4} 2 v | < 2.4 η | {6} 2 v | < 2.4 η | {8} 2 v | < 0.8 η | {p-SP} 2 v | < 0.8 η | {Pb-SP} 2 v v2{LYZ} |η| < 2.4 0 2 4 (GeV/c) T p 0 0.1 0.2 2 v < 150 offline trk N ≤ 120 2 4 (GeV/c) T p 0 0.1 0.2 < 185 offline trk N ≤ 150 2 4 (GeV/c) T p 0 0.1 0.2 < 220 offline trk N ≤ 185 2 4 (GeV/c) T p 0 0.1 0.2 < 260 offline trk N ≤ 220 CMS 2.3 µb-1 (PbPb 2.76 TeV) | < 2.4 η | | > 2} η ∆ {2,| 2 v | < 2.4 η | {4} 2 v | < 2.4 η | {6} 2 v | < 2.4 η | {8} 2 v | < 0.8 η | {SP} 2 v | < 2.4 η | {LYZ} 2 v
Figure 1: (Color online) (Top) The v2 coefficients as a function of pT in pPb collisions for
dif-ferent Ntrkoffline ranges. (Bottom) Same, but for PbPb collisions. The v2{2,|∆η| > 2}and v2{4}
results are from Ref. [38]. For the pPb collisions, the notations p-SP and Pb-SP indicate the pseudorapidity side of the reference event plane, and correspond to the p- and Pb-going direc-tions, respectively. Pseudorapidities are given in the laboratory frame. Systematic uncertainties are indicated by the grey boxes.
The momentum-dependent v2(pT)results in the region|η| < 2.4 for pPb and PbPb collisions
are shown in Fig. 1. The scalar product values, shown separately for the p- and Pb-going event planes, are found to be significantly higher than the multiparticle cumulant (v2{4}, v2{6},
and v2{8}), and Lee–Yang zeros (v2{LYZ}) results. The two-particle correlations (v2{2}) and
lower-order cumulant (v2{4}) measurements shown in the figure are from Ref. [38]. As will
be discussed when presenting the yield-weighted integral v2values, the greater values found
for v2{SP} and v2{2} suggest a significant, and expected, contribution of fluctuations in the
11
In the range of pT < 2 GeV/c there is very little difference between the v2{SP} results
ob-tained with the p- and Pb-going side event planes. However, at higher transverse momenta, the p-going event plane leads to systematically larger values. This behavior suggests that the
nonflow contribution has a larger effect on the high-pT v2 values based on the p-going side
event plane. Monte Carlo simulations using the HIJING event generator support a nonflow
component to the v2 signal that increases almost monotonically with pT. In situations where
both the event plane angle and the Q-vector associated with the region of interest are based on small numbers of particles, the nonflow behavior can be significant. It is also possible that the pT-dependent event-plane decorrelation effects might be different on the Pb- and p-going
sides. 0 2 4 (GeV/c) T p 0.1 0.2 2 v < 150 offline trk N ≤ 120 2 4 (GeV/c) T p 0 0.1 0.2 < 185 offline trk N ≤ 150 2 4 (GeV/c) T p 0 0.1 0.2 < 220 offline trk N ≤ 185 2 4 (GeV/c) T p 0 0.1 0.2 < 260 offline trk N ≤ 220 2 4 (GeV/c) T p 0 0.1 0.2 < 300 offline trk N ≤ 260 0 2 4 (GeV/c) T p 0 0.1 0.2 2 v 2 4 (GeV/c) T p 0 0.1 0.2 2 4 (GeV/c) T p 0 0.1 0.2 2 4 (GeV/c) T p 0 0.1 0.2 2 4 (GeV/c) T p 0 0.1 0.2 CMS 35 nb-1 (pPb 5.02 TeV) < -1.6) c.m. η (-2.0 < = 0} C η {p-SP; 2 v < 2.0) c.m. η (1.6 < = 0} C η {Pb-SP; 2 v < -1.6) c.m. η (-2.0 < } ROI η = C η {p-SP; 2 v < 2.0) c.m. η (1.6 < } ROI η = C η {Pb-SP; 2 v
Figure 2: (Color online) (Top) Comparison of v2(pT) distributions located on the Pb-going
(−2.0 < ηc.m. < −1.6) and p-going (1.6< ηc.m. <2.0) sides of the tracker region, with ηC = 0.
The notations p-SP and Pb-SP indicate the pseudorapidity side of the reference event plane and correspond to the p- and Pb-going directions, respectively. (Bottom) Same, but with ηC =ηROI,
as discussed in the text. Pseudorapidities are given in the laboratory frame. Systematic uncer-tainties are indicated by the grey boxes.
In contrast to Fig. 1, which uses an η region that is symmetric in the lab frame, Fig. 2 compares the v2{SP}(pT)results for symmetric pseudorapidity ranges in the center-of-mass frame. The
laboratory frame results for the range of 2.0<η<2.4 correspond approximately to the
center-of-mass range of 1.6<ηc.m.<2.0 and are obtained with respect to the event plane found on the
Pb-going side with−5.0 < η< −3.0, as indicated with the notation v2{Pb-SP}. Similarly, the
range of−1.6< η< −1.2 approximately corresponds to−2.0<ηc.m.< −1.6. Here the results
are obtained with respect to the event plane found on the p-going side with 3.0 < η < 5.0, as
indicated with the notation v2{p-SP}. The measured values are shown separately with ηC =0
and= ηROI. The reference event plane used in each case corresponds to the more distant HF
detector. In the region with 1.5< pT < 3.0 GeV/c, the enhancement observed on the Pb-going
side (−2.0<ηc.m.< −1.6; p-SP) with ηC =0 (top row) is reduced by taking ηC =ηROI(bottom
row). This dependence on ηCsuggests the presence of event plane decorrelation.
Further evidence for event plane decorrelation is seen by comparing the pseudorapidity de-pendence of the yield-weighted v2 values for 0.3 < pT < 3.0 GeV/c. This is shown in Figs. 3
and 4 for the pPb and PbPb collisions, respectively. The top row in each figure shows the scalar product results with ηC =0 and the bottom row with ηC =ηROI. For the pPb collisions, results
2 − −1 0 1 2 η 0.05 0.1 2 v < 150 offline trk N ≤ 120 < 3.0 GeV/c T 0.3 < p 2 − −1 0 1 2 η 0 0.05 0.1 < 185 offline trk N ≤ 150 2 − −1 0 1 2 η 0 0.05 0.1 < 220 offline trk N ≤ 185 2 − −1 0 1 2 η 0 0.05 0.1 < 260 offline trk N ≤ 220 2 − −1 0 1 2 η 0 0.05 0.1 < 300 offline trk N ≤ 260 2 − −1 0 1 2 η 0 0.05 0.1 2 v 2 − −1 0 1 2 η 0 0.05 0.1 2 − −1 0 1 2 η 0 0.05 0.1 2 − −1 0 1 2 η 0 0.05 0.1 2 − −1 0 1 2 η 0 0.05 0.1 CMS 35 nb-1 (pPb 5.02 TeV) = 0} C η {p-SP; 2 v = 0} C η {Pb-SP; 2 v } ROI η = C η {p-SP; 2 v } ROI η = C η {Pb-SP; 2 v
Figure 3: (Color online) (Top) Yield-weighted v2{SP}with 0.3< pT <3.0 GeV/c as a function of
ηin pPb collisions for different Ntrkofflineranges with ηC =0. (Bottom) Same, but with ηC =ηROI.
The notations p-SP and Pb-SP indicate the pseudorapidity side of the reference event plane and correspond to the p- and Pb-going directions, respectively. Pseudorapidities are given in the laboratory frame. Systematic uncertainties are indicated by the grey boxes.
2 − −1 0 1 2 η 0.05 0.1 2 v < 150 offline trk N ≤ 120 < 3.0 GeV/c T 0.3 < p 2 − −1 0 1 2 η 0 0.05 0.1 < 185 offline trk N ≤ 150 2 − −1 0 1 2 η 0 0.05 0.1 < 220 offline trk N ≤ 185 2 − −1 0 1 2 η 0 0.05 0.1 < 260 offline trk N ≤ 220 2 − −1 0 1 2 η 0 0.05 0.1 < 300 offline trk N ≤ 260 2 − −1 0 1 2 η 0 0.05 0.1 2 v 2 − −1 0 1 2 η 0 0.05 0.1 2 − −1 0 1 2 η 0 0.05 0.1 2 − −1 0 1 2 η 0 0.05 0.1 2 − −1 0 1 2 η 0 0.05 0.1 CMS 2.3 µb-1 (PbPb 2.76 TeV) = 0} C η -SP; + {HF 2 v = 0} C η -SP; -{HF 2 v } ROI η = C η -SP; + {HF 2 v } ROI η = C η -SP; -{HF 2 v
Figure 4: (Color online) (Top) Yield-weighted v2{SP} coefficients as a function of η in PbPb
collisions for different Ntrkofflineranges with ηC = 0. (Bottom) Same, but with ηC = ηROI. The
notations HF+and HF−indicate the pseudorapidity side of the reference event plane. Pseudo-rapidities are given in the laboratory frame. Systematic uncertainties are indicated by the grey boxes.
13
are shown separately over the full pseudorapidity range of the CMS tracker using the HF event planes on the p- and Pb-going side of the collision. For the symmetric PbPb collisions, the
re-sults using the HF+and HF−event planes are shown separately. The yield-weighted elliptic
flow coefficients for PbPb collision are found to be≈20% larger than for pPb collisions. In the absence of decorrelation effects, the choice of ηC = 0 or= ηROI would be expected to result in
similar distributions. In previous PbPb studies [62, 63], taking ηC = 0, the v2(η)values with η < 0 were reported using the event plane with 3.0 < η < 5.0, and the values with η > 0
were reported using the event plane with−5.0< η< −3.0, thus achieving the largest possible
gap in pseudorapidity. Before accounting for an increasing decorrelation of event planes with
an increasing pseudorapidity gap, the v2 values based on p-going and Pb-going side event
planes (pPb collisions) or HF+ and HF− event planes (PbPb collisions) show different
pseu-dorapidity dependences, with the values decreasing as the gap with the reference event plane increases. This reference event plane dependence largely disappears once a correction is ap-plied for decorrelation effects, with the corrected v2values showing very little pseudorapidity
dependence. The resulting boost invariance is consistent with the azimuthal dependence being determined by the initial-state geometry. For the pPb collisions, the results with 2.0< η< 2.4
determined using the p-going side reference event plane are systematically higher in each of the Ntrkoffline ranges. This is consistent with the reduced multiplicity associated with this eta region, allowing for an increased influence of nonflow effects.
The current results suggest that event plane decorrelation effects might be significant in trying to understand the pseudorapidity dependence of the flow coefficients. The results with 2.0 <
η < 2.4 determined using the p-going side reference event plane are systematically higher,
suggesting the possible influence of nonflow effects.
Expanding on the results in Figs. 3 and 4, which show only v2from the scalar product method,
the yield-weighted average v2 values for all of the analysis methods are shown in Fig. 5. It
is interesting to note that the pseudorapidity dependence is almost flat for the scalar product calculations where ηC =ηROI. This is in contrast to the scalar product results for ηC =0 and for
the higher-order particle correlation analyses, where the v2 values at larger pseudorapidities
are significantly smaller. It is only for the scalar product analysis with ηC =ηROI that a partial
accounting for the event plane decorrelation behavior is achieved. Both the cumulant and LYZ analyses employ integral reference flows based on the full range of the CMS tracker and thus are not able to account for decorrelation effects. There is an apparent asymmetry as a function of pseudorapidity for the LYZ results for the two highest Ntrkofflineranges, with a larger v2signal
observed on the Pb-going side event plane. Although this asymmetry appears to be larger than that found for the cumulant or scalar product analyses, the large statistical uncertainties make a direct comparison difficult.
It can be seen from Fig. 5 that the PbPb results for a given Ntrkofflinerange are consistently higher than the corresponding pPb results. This likely reflects the very different collision geometries for the two systems, with the elliptic flow for PbPb collisions being influenced by the lenticular-shaped overlap region developed in non-central collisions of two Pb nuclei. In a later discus-sion, this result will be contrasted with a similar comparison for the v3harmonic.
As already suggested for the pT-dependent results, the difference between the scalar
prod-uct and two-particle correlations results, as compared to the higher-order correlation studies, is likely to reflect initial-state fluctuation effects. Event-by-event fluctuations in the location of the participant nucleons can have a large and method-dependent influence on the harmonic coefficients [72, 73]. Expressing the fluctuations in terms of the azimuthal anisotropy in the par-ticipant plane v, where the harmonic number is suppressed, the magnitude of the fluctuations
2 − −1 0 1 2 η 0 0.05 0.1 2 v < 150 offline trk N ≤ 120 < 3.0 GeV/c T 0.3 < p 2 − −1 0 1 2 η 0 0.05 0.1 < 185 offline trk N ≤ 150 2 − −1 0 1 2 η 0 0.05 0.1 < 220 offline trk N ≤ 185 2 − −1 0 1 2 η 0 0.05 0.1 < 260 offline trk N ≤ 220 CMS 35 nb-1 (pPb 5.02 TeV) = 0} C η {SP; 2 v } ROI η = C η {SP; 2 v {4} 2 v {6} 2 v {8} 2 v {LYZ} 2 v 2 − −1 0 1 2 η 0 0.05 0.1 2 v < 150 offline trk N ≤ 120 < 3.0 GeV/c T 0.3 < p 2 − −1 0 1 2 η 0 0.05 0.1 < 185 offline trk N ≤ 150 2 − −1 0 1 2 η 0 0.05 0.1 < 220 offline trk N ≤ 185 2 − −1 0 1 2 η 0 0.05 0.1 < 260 offline trk N ≤ 220 CMS 2.3 µb-1 (PbPb 2.76 TeV) = 0} C η {SP; 2 v } ROI η = C η {SP; 2 v {4} 2 v {6} 2 v {8} 2 v {LYZ} 2 v
Figure 5: (Color online) (Top) Yield-weighted v2 values calculated using the scalar product,
cumulant, and LYZ methods as a function of η in pPb collisions for different Ntrkoffline ranges. (Bottom) Same, but for PbPb collisions. The v2{SP}results are based on the furthest HF event
plane in pseudorapidity from the particles of interest. Pseudorapidities are given in the labo-ratory frame. Systematic uncertainties are indicated by the grey boxes.
15
is given by σv2≡v2
− hvi2. To leading order in σv[73], two- and four-particle correlations are
affected differently, with
v{2}2=v2
= hvi2+σv2 (21)
and
v{4}2=2v2 2
−Dv4E1/2 ≈ hvi2−σv2. (22)
Multiparticle correlations with more than four particles are expected to give results similar to those of four-particle correlations. Fluctuations affect the scalar product and two-particle correlations in a similar manner. The difference between the scalar product and higher-order cumulant results therefore reflects the initial-state fluctuations.
Using Eqs. (21) and (22), the fluctuation ratio σv/hvican be calculated as
σv hvi = s v2{2}2−v2{4}2 v2{2}2+v2{4}2 = s v2{SP}2−v2{4}2 v2{SP}2+v2{4}2 . (23)
This ratio is shown in Fig. 6 for the pPb and PbPb collisions in different Ntrkoffline ranges. The v2{SP}results with ηC = 0 are used in the calculations since the v2{4} results are expected
to be affected by decorrelation effects. The fluctuation component is found to be significantly larger for the pPb collisions as compared to the PbPb results. A small (15–20%) increase in the ratio is found for both the pPb and PbPb systems as the Ntrkofflinerange increases. The pPb system also shows an increase in the ratio as the pseudorapidity increases.
2 − −1 0 1 2 η 0 0.2 0.4 0.6 0.8 /<v>v σ < 3.0 GeV/c T 0.3 < p = 0 C η < 150 offline trk N ≤ 120 2 − −1 0 1 2 η 0 0.2 0.4 0.6 0.8 offline < 185 trk N ≤ 150 2 − −1 0 1 2 η 0 0.2 0.4 0.6 0.8 offline < 220 trk N ≤ 185 2 − −1 0 1 2 η 0 0.2 0.4 0.6 0.8 offline < 260 trk N ≤ 220 CMS 35 nb-1 (pPb 5.02 TeV); 2.3 µb-1 (PbPb 2.76 TeV) pPb PbPb
Figure 6: (Color online) The ratio σv/hviin the pPb and PbPb systems as a function of
pseu-dorapidity for the indicated Ntrkofflineranges. Pseudorapidities are given in the laboratory frame. Systematic uncertainties are indicated by the grey boxes.
The results presented here can be used to evaluate in more detail previous CMS analyses which suggest a significant pseudorapidity dependence of the v2coefficient of pPb collisions, with a
larger “flow” signal on the Pb-going side [74]. That study was based on a two-particle corre-lation analysis and focused on the ratio v2(η)/v2(η =0). Since the Ref. [74] analysis does not
take into account decorrelation effects, it is most closely related to the scalar product analysis
with ηC = 0 and to the multiparticle correlation measurements based on the integral flow
co-efficients found using an extended range of the CMS tracker acceptance. The Ref. [74] results are compared to the scalar product and four-particle cumulant results in Fig. 7. Agreement is
found among these measurements. The scalar product results with ηC = ηROI, also shown in
Fig. 7, fall off more slowly when moving away from midrapidity.
To explore further the possible asymmetry in the pseudorapidity-dependent v2results of Fig. 5
2 − −1 0 1 2 η 0.4 0.6 0.8 1 1.2 = 0) η ( 2 ) / v η (2 v < 3.0 GeV/c T 0.3 < p < 260 offline trk N ≤ 220 CMS 35 nb-1 (pPb 5.02 TeV) {4} 2 v =0} C η {SP, 2 v } ROI η = C η {SP, 2 v p-trig 2-part Pb-trig 2-part
Figure 7: (Color online) Comparison of the scalar product (v2{SP}) and cumulant (v2{4})
re-sults for the ratio v2(η)/v2(η = 0)with the two-particle correlation results from Ref. [74] for
pPb collisions at √sNN = 5.02 TeV and with 220 ≤ Ntrkoffline < 260. The scalar product results with η < 0 use the p-side reference event plane with 3.0< η<5.0, and the results with η >0
are based on the Pb-side reference event plane with−5.0 < η< −3.0. The two-particle
corre-lation results of Ref. [74] for p-side (p-trig 2-part) and Pb-side (Pb-trig 2-part) trigger particles
are shown without the peripheral v2 component subtraction, a correction for nonflow effects
that increases the v2harmonics. Pseudorapidities are given in the laboratory frame. Error bars
are statistical uncertainties.
0 0.5 1 1.5 2 c.m. η 0.6 0.8 1 η - 2 / v η 2 v < 150 offline trk N ≤ 120 0.5 1 1.5 2 c.m. η 0.6 0.8 1 < 185 offline trk N ≤ 150 0.5 1 1.5 2 c.m. η 0.6 0.8 1 < 220 offline trk N ≤ 185 0.5 1 1.5 2 c.m. η 0.6 0.8 1 < 260 offline trk N ≤ 220 CMS 35 nb-1 (pPb 5.02 TeV) {4} 2 v = 0} C η {SP; 2 v } ROI η = C η {SP; 2 v 2-part
Figure 8: (Color online) Ratio of the p- to Pb-going side v2coefficients at comparable ηc.m.values
for pPb collisions. The two-particle correlation results (labelled “2-part”) are from Ref. [74]. The reference HF event plane is the one furthest from the particles of interest.
17 2 − −1 0 1 2 η 0.02 0.04 3 v < 150 offline trk N ≤ 120 < 3.0 GeV/c T 0.3 < p 2 − −1 0 1 2 η 0 0.02 0.04 < 185 offline trk N ≤ 150 2 − −1 0 1 2 η 0 0.02 0.04 < 220 offline trk N ≤ 185 2 − −1 0 1 2 η 0 0.02 0.04 < 260 offline trk N ≤ 220 2 − −1 0 1 2 η 0 0.02 0.04 < 300 offline trk N ≤ 260 2 − −1 0 1 2 η 0 0.02 0.04 3 v 2 − −1 0 1 2 η 0 0.02 0.04 2 − −1 0 1 2 η 0 0.02 0.04 2 − −1 0 1 2 η 0 0.02 0.04 2 − −1 0 1 2 η 0 0.02 0.04 CMS 35 nb-1 (pPb 5.02 TeV) = 0} C η {p-SP; 3 v = 0} C η {Pb-SP; 3 v } ROI η = C η {p-SP; 3 v } ROI η = C η {Pb-SP; 3 v
Figure 9: (Color online) (Top) The v3values from the scalar product method for pPb collisions
at √sNN = 5.02 TeV with ηC = 0. (Bottom) Same, but with ηC = ηROI. The notations p-SP
and Pb-SP indicate the pseudorapidity side of the reference event plane and correspond to the p- and Pb-going directions, respectively. Pseudorapidities are given in the laboratory frame. Systematic uncertainties are indicated by the grey boxes.
Pb-going sides at comparable center-of-mass pseudorapidity for pPb collisions. The results are shown for the scalar product analyses with ηC = 0 and= ηROI, and for the four-particle
cumulant analysis. Also shown are the comparable results from the Ref. [74] analysis. For the pPb results where decorrelation effects are not taken into account (i.e., v2{SP, ηC = 0}
and v2{4}), the going side values are significantly larger. The asymmetry between the
Pb-going and p-Pb-going sides largely disappears when decorrelation effects are taken into account. A small asymmetry continues to be present when decorrelation effects are considered (i.e., v2{SP, ηC = ηROI}), although it needs to be recognized that the procedure of moving the ηC
range with ηROIis not expected to fully account for these effects if a torque-effect decorrelation
is present; there may be some additional influence of nonflow effects when the η gap between the ηCand either the ηAor ηB event planes becomes small.
In contrast to the second order Fourier coefficients discussed above, triangular flow, corre-sponding to the v3 Fourier harmonic, is believed to arise from fluctuations in the participant
geometry in collisions of heavy nuclei. It is interesting to see how this behavior extends to the very asymmetric pPb system. Fig. 9 shows the scalar product results for the pPb collisions at√sNN = 5.02 TeV with ηC = 0 (top) and= ηROI (bottom), respectively, as a function of η.
Yield-weighted v3 values with 0.3 < pT < 3.0 GeV/c are shown. A pronounced jump in v3,
which becomes smaller with increasing Ntrkoffline, is observed for η > 2 when using the p-going side reference event plane. This could be due to nonflow effects when the ROI is close to the reference event plane. For the Pb-going side reference event plane, a similar, but much smaller effect, may be present when taking ηC =ηROI.
A small pseudorapidity dependence is seen in the v3{ηC = ηROI}results, with the values
be-coming smaller on the p-going side. This might suggest a changing level of fluctuations driving the triangular flow signal. The pseudorapidity dependence appears to become less significant as Ntrkofflineincreases. Fig. 10 shows the corresponding scalar product results for the PbPb colli-sions at√sNN = 2.76 TeV with ηC = 0 (top) and= ηROI (bottom). The v3values are found to
increase with increasing Ntrkofflinefor both systems, as previously observed in Ref. [38]. However, contrary to what is found for the v2 coefficients, the v3values are very similar for the pPb and
PbPb systems in a given Ntrkofflinerange.
2 − −1 0 1 2 η 0.02 0.04 3 v < 150 offline trk N ≤ 120 < 3.0 GeV/c T 0.3 < p 2 − −1 0 1 2 η 0 0.02 0.04 < 185 offline trk N ≤ 150 2 − −1 0 1 2 η 0 0.02 0.04 < 220 offline trk N ≤ 185 2 − −1 0 1 2 η 0 0.02 0.04 < 260 offline trk N ≤ 220 2 − −1 0 1 2 η 0 0.02 0.04 < 300 offline trk N ≤ 260 2 − −1 0 1 2 η 0 0.02 0.04 3 v 2 − −1 0 1 2 η 0 0.02 0.04 2 − −1 0 1 2 η 0 0.02 0.04 2 − −1 0 1 2 η 0 0.02 0.04 2 − −1 0 1 2 η 0 0.02 0.04 CMS 2.3 µb-1 (PbPb 2.76 TeV) = 0} C η -SP; + {HF 3 v = 0} C η -SP; -{HF 3 v } ROI η = C η -SP; + {HF 3 v } ROI η = C η -SP; -{HF 3 v
Figure 10: (Color online) (Top) The v3values from the scalar product method for PbPb collisions
at√sNN = 2.76 TeV with ηC = 0. (Bottom) Same, but with ηC = ηROI. The notations HF+ and
HF−indicate the pseudorapidity side of the reference event plane. Pseudorapidities are given in the laboratory frame. Systematic uncertainties are indicated by the grey boxes.
In order to show the system dependence of v2and v3more directly, Fig. 11 shows scalar product
results with ηC = ηROI for both the pPb and PbPb systems. The v3 values, believed to result
almost entirely from initial geometry fluctuations, are almost the same for the two systems. The v2values are still likely to reflect the lenticular shape of the collision geometry in the PbPb
system, leading to larger v2 coefficients than seen for the pPb system. The PbPb v2 values are
also found to increase with increasing event activity, reflecting the additional contribution of the changing collision overlap geometry.
2 − −1 0 1 2 η 0 0.05 0.1 n v < 3.0 GeV/c T 0.3 < p < 185 offline trk N ≤ 150 2 − −1 0 1 2 η 0 0.05 0.1 < 220 offline trk N ≤ 185 2 − −1 0 1 2 η 0 0.05 0.1 < 260 offline trk N ≤ 220 2 − −1 0 1 2 η 0 0.05 0.1 < 300 offline trk N ≤ 260 CMS 35 nb-1 (pPb 5.02 TeV); 2.3 µb-1 (PbPb 2.76 TeV) pPb } ROI η = C η {SP; 2 v PbPb } ROI η = C η {SP; 2 v pPb } ROI η = C η {SP; 3 v PbPb } ROI η = C η {SP; 3 v
Figure 11: (Color online) The v2 and v3 values for pPb (PbPb) collisions at
√
sNN =
5.02(2.76)TeV with ηC = ηROI. The vn{SP}results are based on the furthest HF event plane
in pseudorapidity. Pseudorapidities are given in the laboratory frame. Systematic uncertain-ties are indicated by the grey boxes.
19
6
Summary
The pseudorapidity and transverse momentum dependencies of the elliptic flow v2
coeffi-cient are presented for pPb collisions at√sNN = 5.02 TeV and for peripheral PbPb collisions at√sNN =2.76 TeV based on scalar product, multiparticle cumulant, and Lee–Yang zeros anal-yses. The data are obtained using the CMS detector. The η dependence of the triangular flow v3coefficient is also presented based on the scalar product analysis. For the first time, pT- and
η-dependent cumulant results are presented based on 6- and 8-particle correlations. The
re-sults provide detailed information for the theoretical understanding of the initial state effect and final state evolution mechanism.
All methods lead to a similar η dependence for the v2 harmonic across the pseudorapidity
range studied. The scalar product results are consistently higher than the corresponding mul-tiparticle correlation behavior, with the v2{4}, v2{6}, v2{8}, and v2{LYZ}having comparable
magnitude. An analysis of fluctuations suggests their greater influence in the system formed in pPb as compared to that in the PbPb collisions. No significant pseudorapidity dependence is found for the fluctuation component, although there is a small increase in the level of the fluctuations with increasing Noffline
trk in both the pPb and PbPb systems. The boost invariance
indicated by the decorrelation-corrected results confirms that the flow signal develops very early in the collision and thus reflects the initial-state geometry.
A method is presented to account for the possible decorrelation of the event plane angle with an increasing η gap between two regions of pseudorapidity. The results suggest that most of the η dependence observed using the different methods might be a consequence of the decorrelation effect. Earlier results exploring the η dependence of elliptic flow in heavy ion collisions may need to be reassessed based on the presence of such decorrelation effects.
Only a small difference is found for the v2coefficients on the Pb- and p-going sides for the pPb
collisions once decorrelation effects are considered. This is in contrast to a previous study, in
which the decorrelation effects were not considered and where a larger v2 value was found
on the Pb-going side. If the decorrelation effects are not considered, as is the case with the
current cumulant, LYZ, and scalar product analysis with ηC = 0, good agreement is found
with the previous results. When decorrelation effects are considered, there appears to be very little longitudinal dependence of the flow coefficients near midrapidity.
The yield-weighted v2results of pPb and PbPb collisions at comparable values of Ntrkofflineshow
a similar η dependence, with the heavier system values being about 20% higher than found
for pPb collisions. No significant difference is observed for the PbPb v3 values as compared
to pPb collisions, suggesting that the v3 results are solely a consequence of fluctuations in the
initial-state participant geometry.
Acknowledgments
We congratulate our colleagues in the CERN accelerator departments for the excellent perfor-mance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we grate-fully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Fi-nally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Aus-tria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria);
CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, and ERDF (Estonia); Academy of Fin-land, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Ger-many); GSRT (Greece); OTKA and NIH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS, RFBR and RAEP (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI and FEDER (Spain); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU and SFFR (Ukraine); STFC (United Kingdom); DOE and NSF (USA).
Individuals have received support from the Marie-Curie program and the European Research Council and Horizon 2020 Grant, contract No. 675440 (European Union); the Leventis Foun-dation; the A. P. Sloan FounFoun-dation; the Alexander von Humboldt FounFoun-dation; the Belgian Fed-eral Science Policy Office; the Fonds pour la Formation `a la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Council of Science and Industrial Research, India; the HOMING PLUS program of the Foun-dation for Polish Science, cofinanced from European Union, Regional Development Fund, the Mobility Plus program of the Ministry of Science and Higher Education, the National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities Research Program by Qatar National Research Fund; the Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia programs cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); the Welch Foundation, contract C-1845; and the Weston Havens Foundation (USA).
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